Median of three uniforms below two thirds

Three independent values are drawn from the uniform distribution on $[0,1]$. Find the probability that the middle value, once they are sorted, is smaller than $\tfrac23$.

$\tfrac{20}{27}$. The median is below $\tfrac23$ exactly when at least two of the three values are, a binomial event with success probability $\tfrac23$: $\big(\tfrac23\big)^3+3\big(\tfrac23\big)^2\tfrac13=\tfrac{8}{27}+\tfrac{12}{27}=\tfrac{20}{27}$.